#7056024

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#44892	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2098 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(14,20)(15,23)(17,19)(21,25)(24,27)(26,28);(2,3)(5,6)(7,11)(8,10)(9,12)(13,14,18,20)(15,24,26,21)(16,19,22,17)(23,25,28,27);(5,7,8,12)(6,9,10,11)(13,15,18,26)(14,21,20,24)(16,27,22,25)(17,23,19,28);(1,3,4,2)(7,12)(9,11)(13,16,18,22)(14,17,20,19)(15,25,26,27)(21,28,24,23);(6,10)(9,11)(13,17,18,19)(14,22,20,16)(15,23,26,28)(21,25,24,27)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=h,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6963220);